座標四角形計算機

The Quadrilateral with Points Calculator takes the four (x, y) vertex coordinates of a quadrilateral and returns its area, perimeter, diagonals, and classification. It is the coordinate-geometry analog of the standard quadrilateral calculators — instead of side lengths and angles as inputs, you supply the four corner positions. This tutorial walks through the formulas the calculator uses (Shoelace + distance), the classification logic, and worked examples.

The two key formulas

1. Area via Shoelace formula

Given vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), (x₄, y₄) in order around the quadrilateral:

Area = ½ × |x₁(y₂−y₄) + x₂(y₃−y₁) + x₃(y₄−y₂) + x₄(y₁−y₃)|

This is the 4-vertex case of the general Shoelace formula.

2. Side lengths via distance formula

Each side's length is the distance between its two endpoint vertices:

|P_i P_{i+1}| = √((x_{i+1}−x_i)² + (y_{i+1}−y_i)²)

Compute this for all 4 consecutive pairs. Sum them for the perimeter.

3. Diagonal lengths

The two diagonals connect opposite corners:

Diagonal 1: from (x₁, y₁) to (x₃, y₃)
Diagonal 2: from (x₂, y₂) to (x₄, y₄)

Each diagonal length follows from the distance formula.

The classification logic

From the side lengths and diagonals, the calculator can identify which type of quadrilateral you have:

• Square: all 4 sides equal AND both diagonals equal.
• Rectangle: opposite sides equal AND both diagonals equal (but not all 4 sides equal).
• Rhombus: all 4 sides equal BUT diagonals unequal.
• Parallelogram (general): opposite sides equal AND diagonals bisect each other.
• Isosceles trapezoid: one pair of sides parallel, the other pair equal, diagonals equal.
• Trapezoid (general): one pair of sides parallel.
• Kite: two pairs of consecutive equal sides.
• Irregular quadrilateral: none of the above.

Worked example 1 — rectangle

Quadrilateral with vertices (0, 0), (5, 0), (5, 3), (0, 3).

Side lengths: 5, 3, 5, 3 — opposite sides equal.
Diagonals: from (0,0) to (5,3) = √(25+9) = √34. From (5,0) to (0,3) = √(25+9) = √34. Equal.
Area = 5 × 3 = 15.

Classification: rectangle (5 × 3 dimensions).

Worked example 2 — rhombus

Vertices (0, 0), (3, 4), (6, 0), (3, −4).

Sides: each is √(9+16) = √25 = 5. All 4 sides equal.
Diagonals: horizontal one (0,0)-(6,0) length 6, vertical one (3,4)-(3,−4) length 8. Unequal.
Area via Shoelace: ½|0(4 − (−4)) + 3(0 − 0) + 6(−4 − 4) + 3(0 − 0)| = ½|0 + 0 − 48 + 0| = 24.

Classification: rhombus (4 equal sides, unequal diagonals).

Worked example 3 — irregular trapezoid

Vertices (0, 0), (6, 0), (5, 4), (1, 4).

Sides: bottom (0,0)-(6,0) = 6, right (6,0)-(5,4) = √(1+16) = √17, top (5,4)-(1,4) = 4, left (1,4)-(0,0) = √(1+16) = √17.

Top and bottom are both horizontal (y = 0 and y = 4) — they're parallel. Right and left have equal length (√17). So this is an isosceles trapezoid.

Order of vertices matters

Like the Shoelace formula, the calculator requires vertices listed in ORDER around the quadrilateral (clockwise or counter-clockwise). Random order creates a self-intersecting "bowtie" shape with wrong area.

For a quadrilateral ABCD, list (x_A, y_A), (x_B, y_B), (x_C, y_C), (x_D, y_D) so that A→B→C→D→A traces the boundary without crossing.

Why coordinate input is useful

Coordinate input is often the natural format when:

• You measured corners with GPS or a tape measure to known reference points.
• You're working in a CAD or graphics program where vertices have explicit coordinates.
• The quadrilateral is irregular and you can't easily classify it without computation.
• You need to verify a quadrilateral's type from coordinate data.

The convex hull check

The calculator assumes the four points form a simple (non-self-intersecting) convex quadrilateral. For concave quadrilaterals (one interior angle exceeds 180°), the calculations still work but the area returned is for the actual quadrilateral as drawn, not its convex hull.

Real-world applications

• Land surveying. Computing the area of a four-cornered land plot from GPS coordinates.
• CAD design. Verifying that a 4-vertex shape has the intended type (rectangle vs general parallelogram, etc.).
• Image processing. Computing properties of detected quadrilateral regions in computer vision.
• Architecture. Working with irregular floor plans defined by corner coordinates.

Common mistakes

• Out-of-order vertices. The most common error. Always list in boundary order (clockwise or counter-clockwise).
• Mixing up which vertex is the "opposite" for the diagonal. Diagonals connect 1-to-3 and 2-to-4 (not adjacent vertices).
• Using straight-line distance for non-Euclidean coordinates. The calculator assumes Cartesian coordinates. GPS lat/lon need separate handling for large distances.
• Forgetting absolute value in area. The Shoelace formula can return negative if vertices are listed clockwise. Area is always positive — take |result|.